This paper is concerned with attempts that have been made to incorporate measures of risk in the selection of financial investments. The methodology that is developed here has been primarily aimed at the evaluation of capital investment projects, but the concepts and ideas are also relevant in the financial investment field. Implicit in all this discussion is the fact that when information is obtained in order to calculate an internal rate of return or a net present value, the figures put into the calculation are estimates rather than precise and exact quantities. In some instances the estimates may be felt to be very good, whilst in other cases it may be felt that there is a wide range of possible deviations. Hence the evaluation of any project or projects should, correctly, be described not by a single criterion but by a range of possible values, some of which are judged to be more likely than others. Having said this, there now comes the need to decide upon the method by which choice is to be exercised. Should it be by the choice of the highest average value of the criterion, or by choosing some more conservative rule?